These conditions are met by many well-used thresholding functions, including the soft thresholding rule \(T_\lambda (z)={\rm sgn}(z)(z-\lambda )_{+}\) , where sgn\((z)\)  is the sign function such that sgn\((z)=1\)  if \(z > 0\) , sgn\((z) = 0\)  if \(z=0\) , and sgn\((z) = -1\)  if \(z < 0\) ,
and the adaptive lasso rule \(T_\lambda (z)=z(1-|\lambda /z|^{\eta })_+\)  with
\(\eta \ge 1\)  (see Rothman et al. [1]}).
The hard thresholding function does not satisfy Condition (1), but our analysis also applies to hard thresholding under similar conditions.
